Calculating Variance and Standard Deviation of Transformed Discrete Random Variables

Addictive data analysis is a key aspect of handling discrete random variables, particularly in understanding the changes and transformations they undergo. This article will guide you through the process of calculating the variance and standard deviation of a transformed discrete random variable when you are given VarX and the transformation equation. We will consider a scenario where Y 2X 7 and VarX 5.

Understanding the Transformation

When dealing with the transformation of a random variable X into another variable Y, it is important to understand the properties of variance and standard deviation. For the transformation Y aX b, where a and b are constants, we can use the following properties:

Var(aX b) a^2 Var(X) -- For the variance SD(aX b) |a| SD(X) -- For the standard deviation

These properties provide a straightforward method to calculate the variance and standard deviation of the transformed variable based on the original variable.

Step-by-Step Calculation

Let's go through the steps to calculate the variance and standard deviation for Y given that Y 2X 7 and VarX 5.

Step 1: Calculate the Variance of Y (VarY)

Given the transformation Y aX b, where a 2, we can find VarY using the property of variance:

VarY a^2 Var(X)

Plugging in the values:

VarY 2^2 Var(X) 4 5 20

Step 2: Calculate the Standard Deviation of Y (SDY)

Once we have the variance, we can find the standard deviation using the square root of the variance:

SDY u221AVarY u221A20 2u221A5

Step 3: Calculate the Standard Deviation of 2Y (SD2Y)

For the standard deviation of the transformed variable 2Y, we use the property of the standard deviation for scaling:

SD2Y a SDY 2 2u221A5 4u221A5

Final Results

Now, let's compile our findings:

VarY 20 SDY 2u221A5 SD2Y 4u221A5

By applying these properties and performing the necessary calculations, we have determined the variance and standard deviation for the transformed random variable Y.

Conclusion

In this article, we have explored the process of calculating the variance and standard deviation for a transformed discrete random variable. Understanding these calculations is essential for working with random variables in various fields of statistics and data analysis. The properties of variance and standard deviation for linear transformations provide a powerful tool for simplifying and solving complex problems involving random variables.

Keywords

Variance of Transformed Variables, Standard Deviation, Discrete Random Variables